The Solar Interior Paul Bushby (Newcastle University) STFC Introductory Course in Solar System Physics Leeds, 06 Sept 2010
General Outline 1. Basic properties of the Sun 2. Solar rotation and large-scale flows 3. The solar cycle 4. Solar dynamo theory 5. Open questions
1. Basic properties of the Sun Solar Radius = Solar Mass = 6.96 10 8 m 2 10 30 kg ( 100 times the Earth s radius) ( 330000 times the Earth s mass) Solar Age = 4.57 10 9 years Right: A recent image from the Solar Dynamics Observatory (AIA at 171 Angstrom, showing the solar atmosphere) http://sdo.gsfc.nasa.gov
1. Basic properties of the Sun (cont.) Radial Structure: Core (r < 0.2 solar radii): The region of nuclear energy production, fusion of Hydrogen into Helium via p-p chain Radiative zone (0.2 < r < 0.7): In this region, the energy is transported radially outwards by radiation Convection zone (r > 0.7): The outer part of the Sun is a region of vigorous convective motions, which efficiently transport energy from the interior to the solar surface (Image credit: http://solarscience.msfc.nasa.gov/images/cutaway.jpg)
1. Basic properties of the Sun (cont.) The Sun is highly stratified (see e.g. Stix 2004 for more details) 1.565 10 7 K(centre)<T <5778K (surface) 1.533 10 5 kg/m 3 (centre) <ρ<2.512 10 4 kg/m 3 (surface) Chemical composition: Mostly Hydrogen (H), Helium (He), but also some heavier elements (C, N, Fe, O... ) Centre: 34% Hydrogen, 64% Helium Surface: 73% Hydrogen, 25% Helium The chemical composition is largely uniform throughout the (well mixed) convection zone
2. Solar rotation and large-scale flows The Sun is rotating differentially. At the surface, the equator is rotating more rapidly than the poles. The (sidereal) period of rotation is approximately 25 days at the equator, 35 days at the poles Right: (Schou et al. 1998) The internal rotation profile of the Sun. Contours of constant angular velocity (as inferred from helioseismology) (see lecture this afternoon)
2. Solar rotation and large-scale flows (cont.) Main features: Almost rigidly-rotating radiative interior (r < 0.7) Differential rotation within most of the convection zone (r > 0.7) is only weakly dependent on radius. The surface differential rotation pattern is almost maintained throughout this region, although note the near-surface shear layer... Thin shear layer at the base of the convection zone - the tachocline Above: (Howe et al. 2000) Differential rotation at 5 different latitudes in the Sun The tachocline probably plays an important role in the solar dynamo (see later), but is poorly understood!
2. Solar rotation and large-scale flows (cont.) Northward Velocity (m s -1 ) Time-dependent meridional flows can be measured at the solar surface. These are (generally) polewards in each hemisphere, (although this is not always the case, see e.g. Haber et al. 2002) 15 10 5 0-5 -10-15 -90-60 -30 0 30 60 90 Latitude Left: The mean surface meridional flow, averaged over 167 solar rotations (Taken from Hathaway & Rightmire 2010) Mass conservation arguments imply that there must be an equatorwards flow somewhere in the solar interior. Stratification implies that this flow will be weaker than the surface flow
2. Solar rotation and large-scale flows (cont.) An equatorwards flow has yet to be seen in observations... Left: (Braun & Fan 1998) The radial distribution of the mean latitudinal flow, as inferred from helioseismological measurements Note: No data for r < 0.84 No evidence for a mean equatorwards flow in the outer layers of the Sun. Suggests that this part of the meridional circulation must be confined to deeper layers (and therefore weaker)
2. Solar rotation and large-scale flows (cont.) Theoretical models (large-scale numerical simulations): The most successful current models are anelastic simulations of the convection zone (no radiative interior, no tachocline) ASH code : Anelastic Spherical Harmonics Right: (Miesch et al. 2008) Projections of the convective radial velocity in a spherical shell at r=0.98 (top) and r=0.95 (bottom) Resolution: 5 10 8 grid points
The meridional circulation is dominated The meridional circulation is dominated by a single cell in by a single cell in eachpoleward hemisphere, poleward flow in the upper convection each hemisphere, with flowwith in the upper convection zone and flow in the lower zone (Fig. 6c). 2 andinequatorward flow in equatorward the lower!convection zone (Fig.convection 6c). 2 5. Enstrophy (!, where w45¼! : <!v)patch shown for a 45! ;zone 45! patch lat(! Fig., where w ¼ : < v) shown for a ; 45 in lat! transitionand between a latitude 30, thepoleward longitude r¼ 0:98 (b) r ¼ 0:85 The color of 30!At tude (10at! Y55, the transitionofbetween equa- poleward and equaatr#a. latitude # and ngitude (a) r )¼and 0:98 R# and at (b)(a) r¼ 0:85 R#R. The color %12 %12 to occurs ablescaled is as inlogarithmically. Fig. 1 but here scaled are (a) 10 flows torward occursr#at. These r $ 0:84Y0:85 R#. These cells extend to showntorward here Rangeslogarithmically. shown are (a) 10Ranges cells extend at r $flows 0:84Y0:85 2. Solar rotation and large-scale flows (cont.) %13 %8 and to 10%8 s%2. 10to%710s%2 s%2. (b) 10 Far left: Angular velocity contours, showing the (mean) solar-like differential rotation Near left: Streamlines for the mean mass flux, showing a clear meridional circulation (from Miesch et al. 2008) al rotation, meridional circulation, and mean temperature averaged over longitudeaveraged and time over (58 days). Angular velocity Fig. 6. Differential rotation, meridional circulation, perturbation and mean temperature perturbation longitude and time (58 shown days). as Angular velocity shown as )a)a afunction of radius for selected latitudes as indicated. Contour levels in (a) are every 10 nhz, and the rotation rate of the coordinate system 2D image and (b) a function of radius for selected latitudes as indicated. Contour levels in (a) are every 10 nhz, and the rotation rate of the coordinate system on the color bar (blackon line). the streamfunction (c)streamfunction represent streamlines of represent the mass flux with redof (black contours) and red blue(black contours) and blue 414 nhz) is indicated the Contours color bar of (black line). Contours"ofinthe " in (c) streamlines the mass flux with %1 22 enting clockwiserepresenting and counterclockwise circulation, respectively. The color table saturates at " ¼ &1:2 ; 10 g s. Characteristic amplitudes %1 22 gray contours) clockwise %1 and counterclockwise circulation, respectively. The color table saturates at " ¼ &1:2 ; 10 g s. Characteristic amplitudes poleward) ¼ s0:95 R#. Contouratlevels for the perturbation (d) are everyperturbation 1 K. %1 R# and 5 m s 20r m ( poleward) at r ¼(equatorward) 0:95 R and 5atmr ¼ s%10:75 (equatorward) r ¼ 0:75 R.temperature Contour levels for the temperature (d) are every 1 K. or hv i are at! High Reynolds number (low viscosity) is important. Turbulent stresses balance angular momentum transport due to meridional flows. Calculations at lower Reynolds numbers tend to give multicellular meridional flows... Imposed entropy gradient at the base of the convection zone also needed - suggests that the tachocline plays an important role # #
3. The solar cycle Sunspots: Prominent dark features that appear at the solar surface Left: A movie showing Galileo s sunspot drawings (summer 1612) (Note that the Sun s rotation axis is at 45 degrees to the vertical) Movie taken from the Galileo Project http://galileo.rice.edu/sci/observations/sunspot_drawings.html
3. The solar cycle (cont.) Sunspot structure: Dark umbral region surrounded by a complex filamentary penumbra Sites of magnetic fields. The vertical field at the centre of a typical spot is of the order of 3000G Above: A recent G-band image from the Big Bear Solar Observatory Cooler than surrounding photosphere due to the suppression of convective heat transport by the magnetic field
3. The solar cycle (cont.) Sunspots often occur in pairs, forming within bipolar active regions Formation of active regions: MDI magnetogram (1/3/04) Strong azimuthal field (somewhere) in the solar interior subject to magnetic buoyancy Loops of magnetic flux rise through the solar convection zone (twisting slightly due to Coriolis force - Joy s law) Where the flux breaks through the surface, it forms a pair of magnetic regions, of opposite polarities Left: Illustrative sketch showing the formation of an active region (Parker 1979)
3. The solar cycle (cont.) The butterfly diagram: An approximate 11 year cycle Sunspots form at mid-latitudes at the start of each cycle. Zones of sunspot activity emerge at progressively lower latitudes as the cycle develops
3. The solar cycle (cont.) Sunspots pairs are almost East-West aligned, following Hale s polarity laws (Hale et al. 1919): The polarity of the leading spots in the Northern hemisphere is always the same and is opposite to that of all the leading spots in the Southern hemisphere. These polarities reverse from one sunspot cycle to the next (Image credit: NASA/MSFC/ David Hathaway)
3. The solar cycle (cont.) The solar cycle is (apparently chaotically) modulated. The Maunder minimum (1650-1720) is an extreme example of modulation. Proxy data suggests that similar grand minima have been a recurrent feature of the solar magnetic activity (with one occurring, on average, every 200 years...) (Taken from http://solarscience.msfc.nasa.gov/images/ssn_yearly.jpg)
3. The solar cycle (cont.) Current predictions suggest that the next cycle will be weaker, but this view has changed significantly over the last 2 years... http://www.swpc.noaa.gov/solarcycle/
3. The solar cycle (cont.) From the sunspot observations we can draw some conclusions about the large-scale magnetic field within the solar interior: It has a strong azimuthal component Its global geometry is predominantly dipolar (the azimuthal field is antisymmetric about the equator) It follows a cyclic pattern with a (full) magnetic period of 22 years It is (apparently) confined to low latitudes, with waves of activity migrating from mid to low latitudes during each cycle It is modulated, showing significant variations from one cycle to the next
4. The solar dynamo Section Outline: 4.1 Basic ideas 4.2 The role of differential rotation 4.3 The role of convection 4.4 Parker s dynamo waves 4.5 Mean-field theory 4.6 Modelling the solar dynamo
4.1 Basic ideas The idea of a solar dynamo was first proposed by Joseph Larmor (right) "How could a rotating body such as the Sun become a magnet?" (1919) Basic idea of a hydromagnetic dynamo: In an electrically-conducting fluid, dynamo action is said to occur if a magnetic field is maintained (against the action of ohmic dissipation) by the inductive motions within the fluid It is believed that the solar cycle is driven by a large-scale dynamo Image credit: http://www-history.mcs.st-and.ac.uk/pictdisplay/larmor.html
4.1 Basic ideas (cont.) In MHD, the magnetic field satisfies the induction equation: B U η B t = (U B)+η 2 B represents the magnetic field vector represents the fluid velocity is the magnetic diffusivity (here assumed constant) If U and L represent a typical velocity and lengthscale (respectively), then the magnetic Reynolds number is defined by Rm = UL η (U B) η 2 B In the solar convection zone: 10 6 <Rm<10 10 ( B = 0) (see e.g. Ossendrijver 2003)
4.2 The role of differential rotation At large Rm, magnetic fields are advected with the flow This implies that any shearing motions tend to deform magnetic fields in the direction of the flow Illustrative sketch: In the case of the Sun (left): Differential rotation will tend to shear out magnetic fields in the azimuthal direction
4.2 The role of differential rotation (cont.) An illustrative (local) Cartesian model: z Cartesian frame of reference: x The x-axis points in the southerly direction The y-axis (not shown) points in the easterly direction The z-axis points radially outwards Consider the following shear flow: U = V (z)ŷ To satisfy the solenoidal constraint for the magnetic field (i.e. zero divergence), use a poloidal-toroidal decomposition: B(x, z, t) =B(x, z, t)ŷ + (A(x, z, t)ŷ )
4.2 The role of differential rotation (cont.) This implies that: A t = η 2 A B t = A x dv dz + η 2 B The shear leads to a source term in the equation for B No source term in the equation for A, which implies that this shear flow is not a dynamo... Various antidynamo theorems show that simple flows cannot be dynamos. Furthermore, many simplified magnetic field configurations cannot be maintained by dynamo action. E.g. Cowling s theorem: Steady, axisymmetric fields cannot be produced by a dynamo
4.3 The role of convection Is there any way of simplifying the dynamo problem? Parker s approach (1955): Convection in a rotating fluid can play a role in the dynamo process. Consider a toroidal magnetic field line: Convective upwellings lead to rising loops of magnetic flux The Coriolis force causes these loops to twist This produces a component of magnetic field in the poloidal direction Illustrative sketch:
4.3 The role of convection (cont.) There will be many rising and twisting loops along any given field line. Averaging over all of these will produce a net regenerative effect, usually referred to as the α-effect In our idealised model, we could interpret A(x, z, t) and as quantities that have been averaged in the y- direction. The α-effect could then be incorporated as follows: B(x, z, t) A t = αb + η 2 A B t = A x dv dz + η 2 B A possible dynamo cycle: B Differential Rotation A α-effect
4.4 Parker s dynamo waves A t = αb + η 2 A Does it work? B t = A x dv dz + η 2 B Make some simplifying assumptions (Parker 1955): α is constant η is constant dv is constant dz Look for one-dimensional solutions of the form: A = Âeσt+ikx B = ˆBe σt+ikx
4.4 Parker s dynamo waves (cont.) Using the given ansatz: σâ = α ˆB ηk 2  σ ˆB = ikv  ηk 2 ˆB V dv dz It is then a simple exercise to eliminate  and ˆB : σ + ηk 2 2 = iαkv = σ + ηk 2 = ± 1 (1 + i) αkv 2 = σ = ηk 2 1 ± (1 + i) D Where the dynamo number is defined by D = αv 2η 2 k 3 Oscillatory solutions: The growth rate, σ, has a positive real part (i.e. growing dynamo waves) if. Negative D implies equatorwards propagation D > 1
4.5 Mean-field theory Parker s model is based on physical intuition. It is possible to derive the α-effect in a more rigorous fashion (see e.g. Moffatt 1978) Recall: B t = (U B)+η 2 B Decompose the magnetic field and the velocity field into mean and fluctuating parts (angled brackets denote an average): B = B 0 + b B 0 = B b = 0 U = U 0 + u U 0 = U u = 0 Note: We assume that the averaging operator satisfies the Reynolds rules (commutes with derivatives etc...)
4.5 Mean-field theory (cont.) Substitute these into the induction equation and average to get: B 0 t = u b + (U 0 B 0 )+η 2 B 0 New Term If the velocity is unaffected by the magnetic field, then it can be shown that this new term is linearly related to the mean field, i.e. u b i = α ij B 0 j β ijk B 0 k x j + higher order terms
4.5 Mean-field theory (cont.) In isotropic turbulence there are no preferred directions of motion and: α ij = αδ ij β ijk = β ijk Using these expressions we obtain the mean-field dynamo equation: B 0 t = (αb 0 + U 0 B 0 )+(η + β) 2 B 0 There are two new effects in mean-field theory: The α-effect: An additional electromotive force, parallel to the mean field The β-effect: Enhanced (turbulent) magnetic diffusivity
4.5 Mean-field theory (cont.) Even in isotropic turbulence, α can only be determined analytically under very special circumstances (e.g. low Rm) What can simulations of convection tell us? Cattaneo & Hughes (2006): A Cartesian model of rotating (incompressible) convection At high Rm, this flow is a good dynamo, but no large-scale field! Right: The α-effect in the upper part of the layer. Strong fluctuations but the time-average is very small... 400 0 400 800 0 0.1 0.2 0.3 0.4 0.5 (a)
4.5 Mean-field theory (cont.) Right: (Brun et al. 2004) B φ ASH code simulation of rotating convection in a spherical shell Again, an efficient dynamo, but large-scale fields are weak compared to small-scale fields B r An inefficient α-effect? Some evidence to suggest that we may need to look beyond a convectively-driven α-effect in mean-field models...
4.5 Mean-field theory (cont.) Other possible sources of an α-effect : Magnetic buoyancy in the presence of rotation (see e.g. Moffatt (1978) A tachocline-based shear instability (Dikpati & Gilman 2001) The decay of bipolar active regions: The loops of magnetic flux connecting the centres of a sunspot pair are usually slightly inclined with respect to the East-West direction. Therefore, the sunspot formation process can act as a surface α-effect Left: A schematic illustration of a surface α-effect (Dikpati & Gilman 2009)
4.6 Modelling the solar dynamo Convection simulations suggest that magnetic flux accumulates in the stably stratified layer below the convection zone Left: (Tobias et al 2001) Simulation of penetrative turbulent convection (a convective layer, overlying a convectively-stable region) acting on a magnetic layer The convective plumes tend to pump magnetic flux into the stable layer The region around the base of the convection zone is also the site of the tachocline, so this region of strong differential rotation plays an important role in most solar dynamo models
4.6 Modelling the solar dynamo (cont.) 1. Tachocline-based dynamos: The dynamo is localised around the base of the convection zone. Azimuthal magnetic field is (mostly) produced by differential rotation in the tachocline Meridional field is regenerated in the lower convection zone, by a convectively-driven α-effect ( interface dynamo ). Alternatively, it is regenerated by some other tachocline-based α-effect (e.g. magnetic buoyancy + rotation) Right: (sketch by Nic Brummell) A schematic illustration of how a tachocline-based dynamo might work
4.6 Modelling the solar dynamo (cont.) Tachocline-based dynamos: Below Left: An analytic fit to the solar differential rotation profile Below Right: Contours of azimuthal magnetic field at the base of the convection zone for a 2D mean-field model
4.6 Modelling the solar dynamo (cont.) Modulated tachocline-based dynamos: By allowing the magnetic field to perturb the differential rotation, it is possible to find parameter values that produce modulated activity cycles, with solar-like grand minima Below Left: A simplified Cartesian model Below Right: A spherical model, with realistic rotation (Beer et al. 1998)
4.6 Modelling the solar dynamo (cont.) 2. Flux transport dynamos: The whole convection zone plays a role in the dynamo Azimuthal magnetic field is (mostly) produced by differential rotation in the tachocline Meridional magnetic field is regenerated at the surface by the decay of active regions (a surface α-effect ) A meridional flow couples the surface to the tachocline. This acts as a conveyor-belt for magnetic flux Right: Analytic fits to the differential rotation and a plausible single-cell meridional circulation (from Dikpati & Charbonneau 1999)
4.6 Modelling the solar dynamo (cont.) Below: Comparing output from a calibrated flux transport model with solar observations (Dikpati & Gilman 2009) No grand minima, otherwise also compares favourably with observations Note: Like all mean-field models, there are many free parameters, that are poorly constrained by theory and observations...
5. Open Questions A selection of open questions for the solar interior: The solar tachocline: Why is is so thin? Is the shear confined to this narrow region by a (primordial?) magnetic field in the solar interior? How did it form? Did it have a similar structure during earlier phases of the solar evolution? etc... (see, e.g., The Solar Tachocline, 2007) Meridional flows: Is the mean flow within the solar convection zone single-cellular or multicellular? How deep is the equatorwards return flow? What are the prospects for measuring this flow?
5. Open Questions (cont.) The solar cycle: To what extent is the solar cycle predicable? Are we due to have another grand minimum soon? The solar dynamo: The nature of the α-effect. Can it be driven by convection in the solar interior, or is some other mechanism required? How effectively do meridional flows couple the surface to the tachocline? Is the solar dynamo a flux transport dynamo or is it an interface dynamo? How can we move beyond mean-field dynamo theory?